Part 1:

Part 1z: Load data and review

brainlang <- read.csv("https://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter16/chap16q15LanguageGreyMatter.csv")
str(brainlang)
## 'data.frame':    22 obs. of  2 variables:
##  $ proficiency: num  0.26 0.44 0.89 1.26 1.69 1.97 1.98 2.24 2.24 2.58 ...
##  $ greymatter : num  -0.07 -0.08 -0.008 -0.009 -0.023 -0.009 -0.036 -0.029 -0.008 -0.023 ...
summary(brainlang)
##   proficiency      greymatter        
##  Min.   :0.260   Min.   :-0.0800000  
##  1st Qu.:1.972   1st Qu.:-0.0195000  
##  Median :2.525   Median :-0.0070000  
##  Mean   :2.400   Mean   : 0.0004091  
##  3rd Qu.:3.163   3rd Qu.: 0.0227500  
##  Max.   :3.850   Max.   : 0.0620000
# vars are proficiency (treatment) and grey matter (outcome)

Part 1a: Display association between the two vars in scatter plot

library(ggplot2)
library(ggpubr)
brainlang_basicviz <- ggplot(data = brainlang,
                             mapping = aes(x = proficiency,
                                           y = greymatter)) +
  geom_point()
brainlang_basicviz

# Adding a more advanced plot that includes R2 value and pearson correlation

brainlang_advancedviz <- ggscatter(brainlang,
                                   x = "proficiency",
                                   y = "greymatter",
                                   add = "reg.line",
                                   conf.int = TRUE,
                                   cor.coef = TRUE,
                                   cor.method = "pearson") +
  labs(title = "Does learning a second language change brain structure?",
       subtitle = "Comparing second language proficiency on grey matter volume",
       x = "Second language proficiency rating",
       y = "Grey matter volume")
brainlang_advancedviz
## `geom_smooth()` using formula 'y ~ x'

Part 1b: Calculate the correlation between second language proficiency and gray-matter density

cor(brainlang$greymatter,
    brainlang$proficiency)
## [1] 0.8183134

Part 1c: SKIP

Part 1d: What are the assumptions when testing the null hypothesis of zero correlation?

library(performance)

brainlang_lm <- lm(greymatter ~ proficiency,
               data = brainlang)

check_model(brainlang_lm)

# Posterior Predictive Check - simulated data compared to the observed data
# Linearity - observation depends on the treatment/condition
# Homogeneity of variance - variances of two or more samples are considered equal
# Independence of errors / Influential Observations - there is no relationship between the errors and the observations
# Normality (of residuals) - treatment follows a normal (Gaussian) distribution

Part 1e: Does the scatter plot support these assumptions? Explain

# Predictions check
check_predictions(brainlang_lm,
                  iterations = 100)

# Predicted lines do not closely fit the observed data

# Linearity check
library(modelr)
## 
## Attaching package: 'modelr'
## The following objects are masked from 'package:performance':
## 
##     mae, mse, rmse
brainlang <- brainlang |>
  add_predictions(brainlang_lm) |>
  add_residuals(brainlang_lm)

ggplot(brainlang,
       mapping = aes(x = pred, y = resid)) +
  geom_point() +
  stat_smooth()
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

# The reference line is not flat / horizontal - data appears non-linear

# Alternative plotting method:
# check_model(brainlang_lm, check = "ncv") %>% 
  # plot()

# Homogeneity:
check_heteroscedasticity(brainlang_lm) |>
  plot()

# again, data appears non-linear

# Normality:
check_normality(brainlang_lm) %>% 
  plot()

# Distribution is close-ish but not that close to the normal curve

# Alternative plotting method:
# check_normality(brainlang_lm) %>% 
# plot(type = "qq")

# Outliers:
plot(brainlang_lm, which = 4)

# distance is larger than previous examples we have covered

check_outliers(brainlang_lm) %>% 
  plot()

# All points are within the contour lines though

Part 1f: Do the results demonstrate that second language proficiency affects gray matter density in the brain? Why or why not?

Notes:

1. Looking at the plots in 1a, the general trend is that as second language profic. increases, grey matter volume increases.
2. I'll check the R2 value to see how closely the independent variable is related to the dependent variable
model_performance(brainlang_lm)
## # Indices of model performance
## 
## AIC      |     BIC |    R2 | R2 (adj.) |  RMSE | Sigma
## ------------------------------------------------------
## -101.941 | -98.668 | 0.670 |     0.653 | 0.021 | 0.022

Notes:

1. R2 value = 0.67. Somewhat strong correlation suggesting that language proficiency does increase grey matter

Part 2:

Part 2z: Load data and review

grass <- read.csv("https://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter17/chap17q19GrasslandNutrientsPlantSpecies.csv")

str(grass)
## 'data.frame':    10 obs. of  2 variables:
##  $ nutrients: int  0 0 0 1 2 3 1 3 4 4
##  $ species  : int  36 36 32 34 33 30 20 23 21 16
summary(grass)
##    nutrients       species     
##  Min.   :0.00   Min.   :16.00  
##  1st Qu.:0.25   1st Qu.:21.50  
##  Median :1.50   Median :31.00  
##  Mean   :1.80   Mean   :28.10  
##  3rd Qu.:3.00   3rd Qu.:33.75  
##  Max.   :4.00   Max.   :36.00

Part 2a: Scatter plot the data

library(ggplot2)

grass_basicviz <- ggplot(data = grass,
                             mapping = aes(x = nutrients,
                                           y = species)) +
  geom_point()
grass_basicviz

Part 2b: What is the rate of change in # plant species supported by nutrient type added?

# Create model
grass_lm <- lm(species ~ nutrients,
                   data = grass)

# Check it's fit
check_model(grass_lm)

# Check the estimate (rate of change) of the fitted line
library(broom)
## 
## Attaching package: 'broom'
## The following object is masked from 'package:modelr':
## 
##     bootstrap
tidy(grass_lm) # estimate = -3.34
## # A tibble: 2 × 5
##   term        estimate std.error statistic    p.value
##   <chr>          <dbl>     <dbl>     <dbl>      <dbl>
## 1 (Intercept)    34.1       2.60     13.1  0.00000108
## 2 nutrients      -3.34      1.10     -3.04 0.0161

Part 2c: Add the least-squared regression line to the scatter plot

library(ggpmisc)
## Loading required package: ggpp
## 
## Attaching package: 'ggpp'
## The following object is masked from 'package:ggplot2':
## 
##     annotate
grass_fit_plot <- ggplot(data = grass_lm,
                         mapping = aes(x = nutrients,
      y = species)) +
  stat_poly_line() +
  stat_poly_eq(aes(label = paste(after_stat(eq.label),
                                 after_stat(rr.label), sep = "*\", \"*"))) +
  geom_point() +
  stat_smooth(method = "lm",
              formula = y ~ x)
grass_fit_plot

### R2 = 0.54

Part 2d: Test the null hypothesis on no treatment effect on the number of plant species

Notes:

1. treatment = nutrients = x
2. outcome = species = y
3. Math:
    a. f(x) = 34.1 - 3.34x
    b. f(0) = 34.1

Part 3:

Part 3z: Load data and review

beetle <- read.csv("https://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter17/chap17q25BeetleWingsAndHorns.csv")

str(beetle)
## 'data.frame':    19 obs. of  2 variables:
##  $ hornSize: num  0.074 0.079 0.019 0.017 0.085 0.081 0.011 0.023 0.005 0.007 ...
##  $ wingMass: num  -42.8 -21.7 -18.8 -16 -12.8 11.6 7.6 1.6 3.7 1.1 ...
summary(beetle)
##     hornSize           wingMass      
##  Min.   :-0.17700   Min.   :-42.800  
##  1st Qu.:-0.03950   1st Qu.: -7.850  
##  Median : 0.00500   Median :  3.700  
##  Mean   :-0.01089   Mean   :  1.316  
##  3rd Qu.: 0.02100   3rd Qu.: 12.300  
##  Max.   : 0.08500   Max.   : 22.200
beetle_basicviz <- ggplot(data = beetle,
                         mapping = aes(x = hornSize,
                                       y = wingMass)) +
  geom_point()
beetle_basicviz

Part 3a: calculate the residuals

beetle_lm <- lm(wingMass ~ hornSize,
                 data = beetle)
beetle_lm
## 
## Call:
## lm(formula = wingMass ~ hornSize, data = beetle)
## 
## Coefficients:
## (Intercept)     hornSize  
##      -0.129     -132.617
summary(beetle_lm)
## 
## Call:
## lm(formula = wingMass ~ hornSize, data = beetle)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -32.857  -8.715   2.157   6.984  22.471 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   -0.129      3.355  -0.038   0.9698   
## hornSize    -132.617     45.146  -2.937   0.0092 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.47 on 17 degrees of freedom
## Multiple R-squared:  0.3367, Adjusted R-squared:  0.2977 
## F-statistic: 8.629 on 1 and 17 DF,  p-value: 0.009201

Part 3b: Produce a residual plot

beetle <- beetle %>% 
  add_predictions(beetle_lm) %>% 
  add_residuals(beetle_lm)

beetle_residual_plot <- ggplot(data = beetle,
                               mapping = aes(x = pred,
                                             y = resid)) +
  geom_point() +
  stat_smooth()
beetle_residual_plot
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Part 3c: evaluate main assumptions of linear regression (using provided plot & resid plot)

check_model(beetle_lm)

# linearity and homogeneity of variance look weird, PPC maybe looks weird

# looking at homogeneity of variance closer
check_heteroskedasticity(beetle_lm) %>% 
  plot()

# looking at linearity closer
check_model(beetle_lm, check = "ncv") %>% 
  plot()

# this could be non-linearity

Part 3d: What steps should be taken?

Notes:

1. Transformation for non-linearity by log transformation

Part 3e: Do any other diagnostics seem weird?

check_normality(beetle_lm) %>% 
  plot()

# distribution does not really fit the normal curve too closely

plot(beetle_lm, which = 4)

check_outliers(beetle_lm) %>% 
  plot(type = "bar")

check_outliers(beetle_lm) %>% 
  plot()

# there are a few data points (1, 6, 19) that are on the cusp of breaking the contour lines

Part 4:

Part 4z: load data and review

teeth <- read.csv("https://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter17/chap17q30NuclearTeeth.csv")

str(teeth)
## 'data.frame':    16 obs. of  2 variables:
##  $ dateOfBirth: num  1986 1984 1990 1988 1990 ...
##  $ deltaC14   : int  89 109 91 127 99 110 123 105 622 262 ...
summary(teeth)
##   dateOfBirth      deltaC14    
##  Min.   :1964   Min.   : 89.0  
##  1st Qu.:1972   1st Qu.:108.0  
##  Median :1984   Median :125.0  
##  Mean   :1980   Mean   :237.4  
##  3rd Qu.:1988   3rd Qu.:370.0  
##  Max.   :1990   Max.   :622.0
teeth_basicviz <- ggplot(data = teeth,
                          mapping = aes(x = deltaC14,
                                        y = dateOfBirth)) +
  geom_point()
teeth_basicviz

Part 4a: what is the slope of the regression line?

teeth_lm <- lm(dateOfBirth ~ deltaC14,
               data = teeth)

teeth_fit_plot <- ggplot(data = teeth_lm,
                         mapping = aes(x = deltaC14,
                                       y = dateOfBirth)) +
  stat_poly_line() +
  stat_poly_eq(aes(label = paste(after_stat(eq.label),
                                 after_stat(rr.label), sep = "*\", \"*"))) +
  geom_point() +
  stat_smooth(method = "lm",
              formula = y ~ x)
teeth_fit_plot

# slope `is -0.0533

Part 4b: Which pair of lines shows the confidence bands? What do they tell us?

Notes:

1. The inner pair of dotted lines show confidence bands
2. These indicate the range that the population parameter is likely to fall within

Part 4c: Which pair of lines shows the prediction interval? What do they tell us?

Notes:

1. The outer pair of dotted lines show the prediction interval
2. These indicate the range that likely contains a new value for a dependent variable given a new observation

Part 4d: Use augment() and geom_ribbon() to reproduce the above plot with data, fit, fit interval, and prediction interval

teeth_model_log <- lm(log(dateOfBirth) ~ deltaC14,
                      data = teeth)

x_frame <- teeth %>% 
  data_grid(deltaC14 = seq_range(deltaC14, n = 100))

teeth_log_curve <- data_grid(teeth,
                             deltaC14 = seq_range(deltaC14, n = 100)) %>% 
  augment(x = teeth_model_log,
          data = .x,
          interval = "confidence") %>% 
  mutate(across(c(.fitted:.upper),
                .fns = exp))
teeth_log_curve
## # A tibble: 100 × 4
##    deltaC14 .fitted .lower .upper
##       <dbl>   <dbl>  <dbl>  <dbl>
##  1     89     1988.  1985.  1990.
##  2     94.4   1987.  1985.  1990.
##  3     99.8   1987.  1985.  1989.
##  4    105.    1987.  1984.  1989.
##  5    111.    1986.  1984.  1989.
##  6    116.    1986.  1984.  1988.
##  7    121.    1986.  1984.  1988.
##  8    127.    1986.  1983.  1988.
##  9    132.    1985.  1983.  1987.
## 10    137.    1985.  1983.  1987.
## # … with 90 more rows
teeth_pred <- augment(x = teeth_lm,
                      newdata = x_frame,
                      interval = "prediction")
teeth_pred
## # A tibble: 100 × 4
##    deltaC14 .fitted .lower .upper
##       <dbl>   <dbl>  <dbl>  <dbl>
##  1     89     1988.  1980.  1995.
##  2     94.4   1987.  1980.  1995.
##  3     99.8   1987.  1979.  1994.
##  4    105.    1987.  1979.  1994.
##  5    111.    1986.  1979.  1994.
##  6    116.    1986.  1979.  1994.
##  7    121.    1986.  1978.  1993.
##  8    127.    1986.  1978.  1993.
##  9    132.    1985.  1978.  1993.
## 10    137.    1985.  1978.  1992.
## # … with 90 more rows
teeth_conf_pred_plot <- teeth_basicviz + 
  geom_line(data = teeth_log_curve,
            mapping = aes(y = .fitted)) +
  geom_ribbon(data = teeth_log_curve,
              mapping = aes(y = .fitted,
                            ymin = .lower,
                            ymax  =.upper),
              alpha = 0.2) +
  geom_line(data = teeth_pred,
            mapping = aes(y = .fitted)) +
  geom_ribbon(data = teeth_pred,
              mapping = aes(y = .fitted,
                            ymin = .lower,
                            ymax  =.upper),
              alpha = 0.2)
teeth_conf_pred_plot

Part 5: Impress Yourself

Part 5a: Fit the deets and bite model from last lab

mosquito <- read.csv("/Users/nathanstrozewski/Documents/Everything/Education/M.S. Biology UMB/Courses/003_F2022/Biological Data Analysis/Homework/Homework #5/Data/17q24DEETMosquiteBites.csv")

mosquito_model <- lm(bites ~ dose,
                     data = mosquito)

## Look at vcov()

vcov(mosquito_model)
##             (Intercept)         dose
## (Intercept)  0.09929780 -0.025986850
## dose        -0.02598685  0.007437073
## Look at mnormt()

library(mnormt)
rmnorm(4, mean = coef(mosquito_model), varcov = vcov(mosquito_model))
##      (Intercept)       dose
## [1,]    4.266033 -0.4929546
## [2,]    3.753990 -0.3949349
## [3,]    4.255887 -0.4820478
## [4,]    3.663541 -0.3276644

Part 5b: Fit simulations

Notes:

1. Using geom_abline(), make a plot that has the following layers:
    a. the data
    b. the lm fit with a CI, and
    c. simulated lines
mosquito_plot <- ggplot(data = mosquito,
                        mapping = aes(x = dose,
                                      y = bites)) +
  geom_point()
mosquito_plot

mosquito_model_log <- lm(log(bites) ~ dose, data = mosquito)
mosquito_log_curve <- data_grid(mosquito,
                                dose = seq_range(dose, n = 100)) %>% 
  augment(x = mosquito_model_log,
          data = .x,
          interval = "confidence") %>% 
  mutate(across(c(.fitted:.upper),
                .fns = exp))
mosquito_log_curve
## # A tibble: 100 × 4
##     dose .fitted .lower .upper
##    <dbl>   <dbl>  <dbl>  <dbl>
##  1  1.4     3.40   2.81   4.11
##  2  1.45    3.37   2.80   4.07
##  3  1.49    3.34   2.78   4.02
##  4  1.54    3.32   2.77   3.97
##  5  1.59    3.29   2.76   3.92
##  6  1.63    3.26   2.74   3.88
##  7  1.68    3.23   2.73   3.83
##  8  1.73    3.20   2.71   3.79
##  9  1.77    3.18   2.70   3.74
## 10  1.82    3.15   2.68   3.70
## # … with 90 more rows
mosquito_sims <- data.frame(rmnorm(100, mean = coef(mosquito_model), varcov = vcov(mosquito_model)))
mosquito_sims
##     X.Intercept.       dose
## 1       4.179555 -0.5010676
## 2       3.944414 -0.4757730
## 3       3.346060 -0.3172197
## 4       4.065938 -0.4743009
## 5       3.947823 -0.3941353
## 6       4.148586 -0.5125126
## 7       3.972883 -0.4251264
## 8       4.259998 -0.4934171
## 9       4.469265 -0.5288779
## 10      3.829872 -0.4074737
## 11      4.334294 -0.4965038
## 12      3.710554 -0.3820394
## 13      3.559839 -0.3427657
## 14      4.219150 -0.4847547
## 15      4.009487 -0.4672119
## 16      3.806797 -0.4031999
## 17      3.975499 -0.3826295
## 18      3.831982 -0.4069850
## 19      3.789073 -0.3490203
## 20      3.750611 -0.4024477
## 21      3.315931 -0.2564231
## 22      4.407929 -0.5280831
## 23      4.377303 -0.5528935
## 24      3.658381 -0.3697326
## 25      4.396758 -0.5897488
## 26      3.734986 -0.3731768
## 27      3.703550 -0.3328067
## 28      3.755076 -0.4171308
## 29      3.236346 -0.2281632
## 30      3.656518 -0.3418796
## 31      3.781250 -0.3770837
## 32      3.955604 -0.4288560
## 33      4.284029 -0.5286386
## 34      3.194729 -0.2278994
## 35      3.020334 -0.1821160
## 36      3.706582 -0.3738850
## 37      3.615785 -0.3635896
## 38      3.766586 -0.3759920
## 39      3.953232 -0.4345951
## 40      3.780146 -0.3272326
## 41      3.732774 -0.3418665
## 42      4.198520 -0.5096941
## 43      4.134507 -0.4512259
## 44      3.910132 -0.4328051
## 45      4.104486 -0.4619389
## 46      4.410975 -0.5491797
## 47      4.186417 -0.4965248
## 48      4.107593 -0.4625967
## 49      3.710886 -0.3807319
## 50      4.139040 -0.4562042
## 51      4.337959 -0.5462528
## 52      4.292496 -0.5773790
## 53      4.122338 -0.4753490
## 54      4.203322 -0.5045451
## 55      4.512315 -0.5565402
## 56      3.957226 -0.4213563
## 57      3.862381 -0.3798935
## 58      4.103151 -0.4388768
## 59      3.894105 -0.4084257
## 60      3.621001 -0.3522379
## 61      3.686992 -0.3498389
## 62      3.510224 -0.3206722
## 63      3.565779 -0.3366080
## 64      3.848933 -0.3678828
## 65      3.568410 -0.3535435
## 66      3.559730 -0.3207468
## 67      3.739270 -0.3951595
## 68      4.573360 -0.6516120
## 69      3.676290 -0.2937165
## 70      3.685993 -0.3851631
## 71      4.086000 -0.4743703
## 72      4.505385 -0.5617108
## 73      4.583815 -0.5634594
## 74      3.690767 -0.3437096
## 75      3.927508 -0.4197293
## 76      3.879901 -0.4059771
## 77      4.087428 -0.4870025
## 78      3.688273 -0.3843254
## 79      3.597398 -0.3323570
## 80      4.160061 -0.4278735
## 81      4.086790 -0.4743768
## 82      3.746937 -0.3664268
## 83      3.785830 -0.3541823
## 84      4.053035 -0.4498136
## 85      3.337257 -0.2887042
## 86      4.320307 -0.5804430
## 87      3.893212 -0.4115677
## 88      3.453204 -0.3235862
## 89      3.521160 -0.2769960
## 90      3.886727 -0.4218223
## 91      4.122523 -0.4844351
## 92      3.920514 -0.3928466
## 93      4.005262 -0.4685748
## 94      3.714949 -0.3199666
## 95      4.028866 -0.4591009
## 96      3.815141 -0.3738136
## 97      3.523394 -0.3003050
## 98      3.759121 -0.3911920
## 99      3.917464 -0.4252375
## 100     3.952978 -0.3944163
mosquito_plot_5b <- mosquito_plot + 
  geom_line(data = mosquito_log_curve,
            mapping = aes(y = .fitted)) +
  geom_ribbon(data = mosquito_log_curve,
              mapping = aes(y = .fitted,
                            ymin = .lower,
                            ymax  =.upper),
              alpha = 0.5) +
  geom_abline(slope = mosquito_sims$dose,
              intercept = mosquito_sims$X.Intercept)
mosquito_plot_5b

Part 5c: Add prediction intervals for simulated lines

summary(mosquito_model_log)
## 
## Call:
## lm(formula = log(bites) ~ dose, data = mosquito)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.72605 -0.16398  0.01497  0.20449  0.47065 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.48157    0.14695  10.082 1.22e-13 ***
## dose        -0.18368    0.04022  -4.567 3.25e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3099 on 50 degrees of freedom
## Multiple R-squared:  0.2944, Adjusted R-squared:  0.2803 
## F-statistic: 20.86 on 1 and 50 DF,  p-value: 3.25e-05
mosquito_predictions <- data.frame(predict(mosquito_model_log, newdata = mosquito_sims, interval = "predict"))
mosquito_predictions 
##          fit       lwr      upr
## 1   1.573610 0.8671219 2.280099
## 2   1.568964 0.8634068 2.274522
## 3   1.539841 0.8400116 2.239670
## 4   1.568694 0.8631904 2.274197
## 5   1.553969 0.8513841 2.256554
## 6   1.575713 0.8688013 2.282624
## 7   1.559661 0.8559539 2.263369
## 8   1.572205 0.8659987 2.278412
## 9   1.578719 0.8712011 2.286236
## 10  1.556419 0.8533518 2.259486
## 11  1.572772 0.8664519 2.279092
## 12  1.551747 0.8495986 2.253896
## 13  1.544533 0.8437937 2.245273
## 14  1.570614 0.8647265 2.276502
## 15  1.567392 0.8621483 2.272635
## 16  1.555634 0.8527215 2.258546
## 17  1.551855 0.8496857 2.254025
## 18  1.556329 0.8532798 2.259379
## 19  1.545682 0.8447189 2.246645
## 20  1.555496 0.8526105 2.258381
## 21  1.528674 0.8309909 2.226356
## 22  1.578573 0.8710846 2.286061
## 23  1.583130 0.8747192 2.291541
## 24  1.549487 0.8477808 2.251192
## 25  1.589900 0.8801101 2.299689
## 26  1.550119 0.8482896 2.251949
## 27  1.542704 0.8423198 2.243088
## 28  1.558193 0.8547756 2.261610
## 29  1.523483 0.8267884 2.220177
## 30  1.544370 0.8436626 2.245078
## 31  1.550837 0.8488667 2.252807
## 32  1.560346 0.8565034 2.264190
## 33  1.578675 0.8711661 2.286184
## 34  1.523434 0.8267491 2.220119
## 35  1.515025 0.8199276 2.210122
## 36  1.550249 0.8483943 2.252104
## 37  1.548358 0.8468730 2.249843
## 38  1.550636 0.8487055 2.252567
## 39  1.561401 0.8573487 2.265453
## 40  1.541680 0.8414946 2.241865
## 41  1.544368 0.8436606 2.245075
## 42  1.575195 0.8683878 2.282002
## 43  1.564455 0.8597969 2.269114
## 44  1.561072 0.8570851 2.265059
## 45  1.566423 0.8613729 2.271474
## 46  1.582448 0.8741755 2.290720
## 47  1.572776 0.8664550 2.279097
## 48  1.566544 0.8614696 2.271618
## 49  1.551507 0.8494055 2.253608
## 50  1.565370 0.8605294 2.270210
## 51  1.581910 0.8737468 2.290074
## 52  1.587628 0.8783018 2.296953
## 53  1.568886 0.8633444 2.274428
## 54  1.574249 0.8676323 2.280866
## 55  1.583800 0.8752531 2.292347
## 56  1.558969 0.8553984 2.262539
## 57  1.551353 0.8492817 2.253424
## 58  1.562187 0.8579792 2.266395
## 59  1.556594 0.8534922 2.259695
## 60  1.546273 0.8451948 2.247351
## 61  1.545832 0.8448400 2.246825
## 62  1.540475 0.8405230 2.240427
## 63  1.543402 0.8428825 2.243922
## 64  1.549147 0.8475075 2.250786
## 65  1.546513 0.8453879 2.247638
## 66  1.540489 0.8405341 2.240443
## 67  1.554157 0.8515353 2.256779
## 68  1.601263 0.8891370 2.313389
## 69  1.535524 0.8365276 2.234520
## 70  1.552321 0.8500598 2.254582
## 71  1.568707 0.8632006 2.274213
## 72  1.584750 0.8760098 2.293489
## 73  1.585071 0.8762657 2.293876
## 74  1.544707 0.8439333 2.245480
## 75  1.558670 0.8551586 2.262182
## 76  1.556144 0.8531311 2.259157
## 77  1.571027 0.8650567 2.276997
## 78  1.552167 0.8499361 2.254398
## 79  1.542621 0.8422533 2.242989
## 80  1.560166 0.8563587 2.263973
## 81  1.568708 0.8632016 2.274214
## 82  1.548879 0.8472923 2.250466
## 83  1.546630 0.8454823 2.247778
## 84  1.564196 0.8595891 2.268803
## 85  1.534603 0.8357841 2.233422
## 86  1.588190 0.8787499 2.297631
## 87  1.557171 0.8539555 2.260386
## 88  1.541010 0.8409546 2.241066
## 89  1.532452 0.8340466 2.230858
## 90  1.559055 0.8554671 2.262642
## 91  1.570555 0.8646795 2.276431
## 92  1.553732 0.8511940 2.256270
## 93  1.567642 0.8623487 2.272936
## 94  1.540345 0.8404185 2.240272
## 95  1.565902 0.8609555 2.270848
## 96  1.550236 0.8483837 2.252089
## 97  1.536734 0.8375047 2.235963
## 98  1.553428 0.8509498 2.255907
## 99  1.559682 0.8559703 2.263393
## 100 1.554020 0.8514256 2.256615
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
mosquito_predictions_withbounds <- cbind(mosquito_sims, mosquito_predictions)
mosquito_predictions_withbounds
##     X.Intercept.       dose      fit       lwr      upr
## 1       4.179555 -0.5010676 1.573610 0.8671219 2.280099
## 2       3.944414 -0.4757730 1.568964 0.8634068 2.274522
## 3       3.346060 -0.3172197 1.539841 0.8400116 2.239670
## 4       4.065938 -0.4743009 1.568694 0.8631904 2.274197
## 5       3.947823 -0.3941353 1.553969 0.8513841 2.256554
## 6       4.148586 -0.5125126 1.575713 0.8688013 2.282624
## 7       3.972883 -0.4251264 1.559661 0.8559539 2.263369
## 8       4.259998 -0.4934171 1.572205 0.8659987 2.278412
## 9       4.469265 -0.5288779 1.578719 0.8712011 2.286236
## 10      3.829872 -0.4074737 1.556419 0.8533518 2.259486
## 11      4.334294 -0.4965038 1.572772 0.8664519 2.279092
## 12      3.710554 -0.3820394 1.551747 0.8495986 2.253896
## 13      3.559839 -0.3427657 1.544533 0.8437937 2.245273
## 14      4.219150 -0.4847547 1.570614 0.8647265 2.276502
## 15      4.009487 -0.4672119 1.567392 0.8621483 2.272635
## 16      3.806797 -0.4031999 1.555634 0.8527215 2.258546
## 17      3.975499 -0.3826295 1.551855 0.8496857 2.254025
## 18      3.831982 -0.4069850 1.556329 0.8532798 2.259379
## 19      3.789073 -0.3490203 1.545682 0.8447189 2.246645
## 20      3.750611 -0.4024477 1.555496 0.8526105 2.258381
## 21      3.315931 -0.2564231 1.528674 0.8309909 2.226356
## 22      4.407929 -0.5280831 1.578573 0.8710846 2.286061
## 23      4.377303 -0.5528935 1.583130 0.8747192 2.291541
## 24      3.658381 -0.3697326 1.549487 0.8477808 2.251192
## 25      4.396758 -0.5897488 1.589900 0.8801101 2.299689
## 26      3.734986 -0.3731768 1.550119 0.8482896 2.251949
## 27      3.703550 -0.3328067 1.542704 0.8423198 2.243088
## 28      3.755076 -0.4171308 1.558193 0.8547756 2.261610
## 29      3.236346 -0.2281632 1.523483 0.8267884 2.220177
## 30      3.656518 -0.3418796 1.544370 0.8436626 2.245078
## 31      3.781250 -0.3770837 1.550837 0.8488667 2.252807
## 32      3.955604 -0.4288560 1.560346 0.8565034 2.264190
## 33      4.284029 -0.5286386 1.578675 0.8711661 2.286184
## 34      3.194729 -0.2278994 1.523434 0.8267491 2.220119
## 35      3.020334 -0.1821160 1.515025 0.8199276 2.210122
## 36      3.706582 -0.3738850 1.550249 0.8483943 2.252104
## 37      3.615785 -0.3635896 1.548358 0.8468730 2.249843
## 38      3.766586 -0.3759920 1.550636 0.8487055 2.252567
## 39      3.953232 -0.4345951 1.561401 0.8573487 2.265453
## 40      3.780146 -0.3272326 1.541680 0.8414946 2.241865
## 41      3.732774 -0.3418665 1.544368 0.8436606 2.245075
## 42      4.198520 -0.5096941 1.575195 0.8683878 2.282002
## 43      4.134507 -0.4512259 1.564455 0.8597969 2.269114
## 44      3.910132 -0.4328051 1.561072 0.8570851 2.265059
## 45      4.104486 -0.4619389 1.566423 0.8613729 2.271474
## 46      4.410975 -0.5491797 1.582448 0.8741755 2.290720
## 47      4.186417 -0.4965248 1.572776 0.8664550 2.279097
## 48      4.107593 -0.4625967 1.566544 0.8614696 2.271618
## 49      3.710886 -0.3807319 1.551507 0.8494055 2.253608
## 50      4.139040 -0.4562042 1.565370 0.8605294 2.270210
## 51      4.337959 -0.5462528 1.581910 0.8737468 2.290074
## 52      4.292496 -0.5773790 1.587628 0.8783018 2.296953
## 53      4.122338 -0.4753490 1.568886 0.8633444 2.274428
## 54      4.203322 -0.5045451 1.574249 0.8676323 2.280866
## 55      4.512315 -0.5565402 1.583800 0.8752531 2.292347
## 56      3.957226 -0.4213563 1.558969 0.8553984 2.262539
## 57      3.862381 -0.3798935 1.551353 0.8492817 2.253424
## 58      4.103151 -0.4388768 1.562187 0.8579792 2.266395
## 59      3.894105 -0.4084257 1.556594 0.8534922 2.259695
## 60      3.621001 -0.3522379 1.546273 0.8451948 2.247351
## 61      3.686992 -0.3498389 1.545832 0.8448400 2.246825
## 62      3.510224 -0.3206722 1.540475 0.8405230 2.240427
## 63      3.565779 -0.3366080 1.543402 0.8428825 2.243922
## 64      3.848933 -0.3678828 1.549147 0.8475075 2.250786
## 65      3.568410 -0.3535435 1.546513 0.8453879 2.247638
## 66      3.559730 -0.3207468 1.540489 0.8405341 2.240443
## 67      3.739270 -0.3951595 1.554157 0.8515353 2.256779
## 68      4.573360 -0.6516120 1.601263 0.8891370 2.313389
## 69      3.676290 -0.2937165 1.535524 0.8365276 2.234520
## 70      3.685993 -0.3851631 1.552321 0.8500598 2.254582
## 71      4.086000 -0.4743703 1.568707 0.8632006 2.274213
## 72      4.505385 -0.5617108 1.584750 0.8760098 2.293489
## 73      4.583815 -0.5634594 1.585071 0.8762657 2.293876
## 74      3.690767 -0.3437096 1.544707 0.8439333 2.245480
## 75      3.927508 -0.4197293 1.558670 0.8551586 2.262182
## 76      3.879901 -0.4059771 1.556144 0.8531311 2.259157
## 77      4.087428 -0.4870025 1.571027 0.8650567 2.276997
## 78      3.688273 -0.3843254 1.552167 0.8499361 2.254398
## 79      3.597398 -0.3323570 1.542621 0.8422533 2.242989
## 80      4.160061 -0.4278735 1.560166 0.8563587 2.263973
## 81      4.086790 -0.4743768 1.568708 0.8632016 2.274214
## 82      3.746937 -0.3664268 1.548879 0.8472923 2.250466
## 83      3.785830 -0.3541823 1.546630 0.8454823 2.247778
## 84      4.053035 -0.4498136 1.564196 0.8595891 2.268803
## 85      3.337257 -0.2887042 1.534603 0.8357841 2.233422
## 86      4.320307 -0.5804430 1.588190 0.8787499 2.297631
## 87      3.893212 -0.4115677 1.557171 0.8539555 2.260386
## 88      3.453204 -0.3235862 1.541010 0.8409546 2.241066
## 89      3.521160 -0.2769960 1.532452 0.8340466 2.230858
## 90      3.886727 -0.4218223 1.559055 0.8554671 2.262642
## 91      4.122523 -0.4844351 1.570555 0.8646795 2.276431
## 92      3.920514 -0.3928466 1.553732 0.8511940 2.256270
## 93      4.005262 -0.4685748 1.567642 0.8623487 2.272936
## 94      3.714949 -0.3199666 1.540345 0.8404185 2.240272
## 95      4.028866 -0.4591009 1.565902 0.8609555 2.270848
## 96      3.815141 -0.3738136 1.550236 0.8483837 2.252089
## 97      3.523394 -0.3003050 1.536734 0.8375047 2.235963
## 98      3.759121 -0.3911920 1.553428 0.8509498 2.255907
## 99      3.917464 -0.4252375 1.559682 0.8559703 2.263393
## 100     3.952978 -0.3944163 1.554020 0.8514256 2.256615
mosquito_predictions_final <- data.frame(mosquito_predictions_withbounds) %>% 
  mutate(X.Intercept_lwr = X.Intercept. - lwr) %>% 
  mutate(X.Intercept_upr = X.Intercept. + upr)
mosquito_predictions_final
##     X.Intercept.       dose      fit       lwr      upr X.Intercept_lwr
## 1       4.179555 -0.5010676 1.573610 0.8671219 2.280099        3.312433
## 2       3.944414 -0.4757730 1.568964 0.8634068 2.274522        3.081007
## 3       3.346060 -0.3172197 1.539841 0.8400116 2.239670        2.506049
## 4       4.065938 -0.4743009 1.568694 0.8631904 2.274197        3.202747
## 5       3.947823 -0.3941353 1.553969 0.8513841 2.256554        3.096439
## 6       4.148586 -0.5125126 1.575713 0.8688013 2.282624        3.279785
## 7       3.972883 -0.4251264 1.559661 0.8559539 2.263369        3.116929
## 8       4.259998 -0.4934171 1.572205 0.8659987 2.278412        3.394000
## 9       4.469265 -0.5288779 1.578719 0.8712011 2.286236        3.598064
## 10      3.829872 -0.4074737 1.556419 0.8533518 2.259486        2.976520
## 11      4.334294 -0.4965038 1.572772 0.8664519 2.279092        3.467842
## 12      3.710554 -0.3820394 1.551747 0.8495986 2.253896        2.860956
## 13      3.559839 -0.3427657 1.544533 0.8437937 2.245273        2.716045
## 14      4.219150 -0.4847547 1.570614 0.8647265 2.276502        3.354424
## 15      4.009487 -0.4672119 1.567392 0.8621483 2.272635        3.147339
## 16      3.806797 -0.4031999 1.555634 0.8527215 2.258546        2.954076
## 17      3.975499 -0.3826295 1.551855 0.8496857 2.254025        3.125813
## 18      3.831982 -0.4069850 1.556329 0.8532798 2.259379        2.978702
## 19      3.789073 -0.3490203 1.545682 0.8447189 2.246645        2.944355
## 20      3.750611 -0.4024477 1.555496 0.8526105 2.258381        2.898001
## 21      3.315931 -0.2564231 1.528674 0.8309909 2.226356        2.484940
## 22      4.407929 -0.5280831 1.578573 0.8710846 2.286061        3.536844
## 23      4.377303 -0.5528935 1.583130 0.8747192 2.291541        3.502583
## 24      3.658381 -0.3697326 1.549487 0.8477808 2.251192        2.810601
## 25      4.396758 -0.5897488 1.589900 0.8801101 2.299689        3.516648
## 26      3.734986 -0.3731768 1.550119 0.8482896 2.251949        2.886697
## 27      3.703550 -0.3328067 1.542704 0.8423198 2.243088        2.861230
## 28      3.755076 -0.4171308 1.558193 0.8547756 2.261610        2.900301
## 29      3.236346 -0.2281632 1.523483 0.8267884 2.220177        2.409558
## 30      3.656518 -0.3418796 1.544370 0.8436626 2.245078        2.812855
## 31      3.781250 -0.3770837 1.550837 0.8488667 2.252807        2.932383
## 32      3.955604 -0.4288560 1.560346 0.8565034 2.264190        3.099101
## 33      4.284029 -0.5286386 1.578675 0.8711661 2.286184        3.412863
## 34      3.194729 -0.2278994 1.523434 0.8267491 2.220119        2.367980
## 35      3.020334 -0.1821160 1.515025 0.8199276 2.210122        2.200406
## 36      3.706582 -0.3738850 1.550249 0.8483943 2.252104        2.858188
## 37      3.615785 -0.3635896 1.548358 0.8468730 2.249843        2.768912
## 38      3.766586 -0.3759920 1.550636 0.8487055 2.252567        2.917880
## 39      3.953232 -0.4345951 1.561401 0.8573487 2.265453        3.095883
## 40      3.780146 -0.3272326 1.541680 0.8414946 2.241865        2.938652
## 41      3.732774 -0.3418665 1.544368 0.8436606 2.245075        2.889114
## 42      4.198520 -0.5096941 1.575195 0.8683878 2.282002        3.330132
## 43      4.134507 -0.4512259 1.564455 0.8597969 2.269114        3.274710
## 44      3.910132 -0.4328051 1.561072 0.8570851 2.265059        3.053047
## 45      4.104486 -0.4619389 1.566423 0.8613729 2.271474        3.243113
## 46      4.410975 -0.5491797 1.582448 0.8741755 2.290720        3.536800
## 47      4.186417 -0.4965248 1.572776 0.8664550 2.279097        3.319962
## 48      4.107593 -0.4625967 1.566544 0.8614696 2.271618        3.246123
## 49      3.710886 -0.3807319 1.551507 0.8494055 2.253608        2.861481
## 50      4.139040 -0.4562042 1.565370 0.8605294 2.270210        3.278511
## 51      4.337959 -0.5462528 1.581910 0.8737468 2.290074        3.464212
## 52      4.292496 -0.5773790 1.587628 0.8783018 2.296953        3.414194
## 53      4.122338 -0.4753490 1.568886 0.8633444 2.274428        3.258994
## 54      4.203322 -0.5045451 1.574249 0.8676323 2.280866        3.335689
## 55      4.512315 -0.5565402 1.583800 0.8752531 2.292347        3.637062
## 56      3.957226 -0.4213563 1.558969 0.8553984 2.262539        3.101828
## 57      3.862381 -0.3798935 1.551353 0.8492817 2.253424        3.013099
## 58      4.103151 -0.4388768 1.562187 0.8579792 2.266395        3.245172
## 59      3.894105 -0.4084257 1.556594 0.8534922 2.259695        3.040613
## 60      3.621001 -0.3522379 1.546273 0.8451948 2.247351        2.775806
## 61      3.686992 -0.3498389 1.545832 0.8448400 2.246825        2.842152
## 62      3.510224 -0.3206722 1.540475 0.8405230 2.240427        2.669701
## 63      3.565779 -0.3366080 1.543402 0.8428825 2.243922        2.722896
## 64      3.848933 -0.3678828 1.549147 0.8475075 2.250786        3.001426
## 65      3.568410 -0.3535435 1.546513 0.8453879 2.247638        2.723022
## 66      3.559730 -0.3207468 1.540489 0.8405341 2.240443        2.719195
## 67      3.739270 -0.3951595 1.554157 0.8515353 2.256779        2.887735
## 68      4.573360 -0.6516120 1.601263 0.8891370 2.313389        3.684223
## 69      3.676290 -0.2937165 1.535524 0.8365276 2.234520        2.839762
## 70      3.685993 -0.3851631 1.552321 0.8500598 2.254582        2.835933
## 71      4.086000 -0.4743703 1.568707 0.8632006 2.274213        3.222799
## 72      4.505385 -0.5617108 1.584750 0.8760098 2.293489        3.629375
## 73      4.583815 -0.5634594 1.585071 0.8762657 2.293876        3.707549
## 74      3.690767 -0.3437096 1.544707 0.8439333 2.245480        2.846834
## 75      3.927508 -0.4197293 1.558670 0.8551586 2.262182        3.072349
## 76      3.879901 -0.4059771 1.556144 0.8531311 2.259157        3.026770
## 77      4.087428 -0.4870025 1.571027 0.8650567 2.276997        3.222371
## 78      3.688273 -0.3843254 1.552167 0.8499361 2.254398        2.838337
## 79      3.597398 -0.3323570 1.542621 0.8422533 2.242989        2.755145
## 80      4.160061 -0.4278735 1.560166 0.8563587 2.263973        3.303702
## 81      4.086790 -0.4743768 1.568708 0.8632016 2.274214        3.223589
## 82      3.746937 -0.3664268 1.548879 0.8472923 2.250466        2.899645
## 83      3.785830 -0.3541823 1.546630 0.8454823 2.247778        2.940348
## 84      4.053035 -0.4498136 1.564196 0.8595891 2.268803        3.193445
## 85      3.337257 -0.2887042 1.534603 0.8357841 2.233422        2.501473
## 86      4.320307 -0.5804430 1.588190 0.8787499 2.297631        3.441558
## 87      3.893212 -0.4115677 1.557171 0.8539555 2.260386        3.039257
## 88      3.453204 -0.3235862 1.541010 0.8409546 2.241066        2.612249
## 89      3.521160 -0.2769960 1.532452 0.8340466 2.230858        2.687113
## 90      3.886727 -0.4218223 1.559055 0.8554671 2.262642        3.031260
## 91      4.122523 -0.4844351 1.570555 0.8646795 2.276431        3.257844
## 92      3.920514 -0.3928466 1.553732 0.8511940 2.256270        3.069320
## 93      4.005262 -0.4685748 1.567642 0.8623487 2.272936        3.142914
## 94      3.714949 -0.3199666 1.540345 0.8404185 2.240272        2.874531
## 95      4.028866 -0.4591009 1.565902 0.8609555 2.270848        3.167911
## 96      3.815141 -0.3738136 1.550236 0.8483837 2.252089        2.966757
## 97      3.523394 -0.3003050 1.536734 0.8375047 2.235963        2.685889
## 98      3.759121 -0.3911920 1.553428 0.8509498 2.255907        2.908171
## 99      3.917464 -0.4252375 1.559682 0.8559703 2.263393        3.061494
## 100     3.952978 -0.3944163 1.554020 0.8514256 2.256615        3.101553
##     X.Intercept_upr
## 1          6.459654
## 2          6.218936
## 3          5.585730
## 4          6.340135
## 5          6.204377
## 6          6.431210
## 7          6.236252
## 8          6.538410
## 9          6.755502
## 10         6.089358
## 11         6.613387
## 12         5.964450
## 13         5.805112
## 14         6.495652
## 15         6.282123
## 16         6.065344
## 17         6.229524
## 18         6.091361
## 19         6.035719
## 20         6.008992
## 21         5.542288
## 22         6.693990
## 23         6.668843
## 24         5.909574
## 25         6.696447
## 26         5.986935
## 27         5.946638
## 28         6.016686
## 29         5.456523
## 30         5.901596
## 31         6.034056
## 32         6.219794
## 33         6.570213
## 34         5.414848
## 35         5.230456
## 36         5.958686
## 37         5.865628
## 38         6.019153
## 39         6.218684
## 40         6.022012
## 41         5.977850
## 42         6.480522
## 43         6.403621
## 44         6.175191
## 45         6.375959
## 46         6.701695
## 47         6.465514
## 48         6.379211
## 49         5.964494
## 50         6.409251
## 51         6.628032
## 52         6.589449
## 53         6.396766
## 54         6.484188
## 55         6.804662
## 56         6.219766
## 57         6.115805
## 58         6.369546
## 59         6.153801
## 60         5.868352
## 61         5.933817
## 62         5.750651
## 63         5.809700
## 64         6.099720
## 65         5.816047
## 66         5.800173
## 67         5.996049
## 68         6.886748
## 69         5.910809
## 70         5.940575
## 71         6.360213
## 72         6.798875
## 73         6.877691
## 74         5.936247
## 75         6.189689
## 76         6.139058
## 77         6.364425
## 78         5.942671
## 79         5.840387
## 80         6.424034
## 81         6.361004
## 82         5.997404
## 83         6.033608
## 84         6.321838
## 85         5.570679
## 86         6.617938
## 87         6.153599
## 88         5.694269
## 89         5.752018
## 90         6.149369
## 91         6.398954
## 92         6.176784
## 93         6.278198
## 94         5.955222
## 95         6.299715
## 96         6.067229
## 97         5.759357
## 98         6.015027
## 99         6.180858
## 100        6.209594
mosquito_plot_5c <- mosquito_plot + 
  geom_line(data = mosquito_log_curve,
            mapping = aes(y = .fitted)) +
  geom_ribbon(data = mosquito_log_curve,
              mapping = aes(y = .fitted,
                            ymin = .lower,
                            ymax  =.upper),
              alpha = 0.5) +
  geom_abline(slope = mosquito_sims$dose,
              intercept = mosquito_sims$X.Intercept) +
  geom_ribbon(data = mosquito_predictions_final,
              mapping = aes(y = X.Intercept.,
                            ymin = X.Intercept_lwr,
                            ymax = X.Intercept_upr))
mosquito_plot_5c

### Notes: 1. I can’t figure out how to get the ribbon around the geom_abline values 2. Close-ish, maybe! I gave it my best shot

Part 6: Meta Questions

Meta 1: How well do you feel you understand the assumption testing behind a linear model? If there are elements that confuse you, what are they? Why?

Notes:

1. I feel somewhat confident about assumption testing now
2. When we first covered it in lecture, I didn't understand what the assumptions were or how they fit into statistical analysis
3. After additional reading and practice, I get why we look at them and how they help
4. I am still a little confused on

Meta 2: What concepts of linear regression are the clearest for you? Which are the most opaque?

Notes:

1. Outliers makes the most sense - very straightforward
2. The positive predictor check is really cool and makes sense
3. I don't fully get homogeneity of variance, but I am getting there

Meta 3: Even if you did not do the IYKYK part of this assignment do you see how simulation can be used with a fit model? Do you feel it would be useful? Do you have gaps in your understanding about why simulation could work here?

Notes:

1. I attempted IYb and could not get the ribbon of predicted bounds around my geom_abline
2. But going through IYa and some of IYb was helpful to understand HOW we can use simulation with a fit model
3. This would absolutely be useful - I really want to figure out how to finish this so that I can see
4. No conceptual gaps yet - just technical (code) gaps

Meta 4: How much time did this take you, roughly?

Notes:

1. This took me approximately two hours

Meta 5: Please give yourself a weak/sufficient/strong assessment on this assignment

Notes:

1. I would rate myself between sufficient and strong - maybe closer to strong
2. I feel that I understand the concepts covered (mostly)
3. I was able to complete all of the problems, except IYb which I got close-ish on
4. I am definitely getting better at applying my knowledge and problem solving within R